Evaluating and managing credit risks with respect to publicly traded companies are important capabilities for financial service providers. Credit risk analysis is used to evaluate a multitude of daily-occurring financial transactions including, for example: the origination of bank loans, the buying and selling of bonds, transactions in the credit derivatives market, the extension of payment to customers, portfolio development and management and portfolio valuation.
The use of Black-Scholes-based analysis tools to estimate credit risk is known in the art. The well known Black-Scholes algorithm uses as inputs values relating to a particular publicly-traded stock, values relating to the financial market as whole and values relating to a proposed stock option in order to calculate a strike price for an option on the particular stock.
With particular changes to the input variables and the formula of the Black-Scholes algorithm, Robert Merton proposed one of the earliest and simplest default credit models. Merton's model regards the equity in a company as an option on the assets with a strike price equal to the face value of the debt. Merton assumed that the value of the company is governed by a risk-neutralized lognormal stochastic differential equation (SDE) of the form:
                                          d            ⁢                                                  ⁢            V                    V                =                              r            ⁢                                                  ⁢            d            ⁢                                                  ⁢            t                    +                      σ            ⁢                                                  ⁢            d            ⁢                                                  ⁢            W                                              (        1        )            
Here V is the value of the company, r is the risk neutral rate, σ is the company's volatility, and W is the standard Wiener process, the well-know building block continuous-time stochastic process for constructing random models over continuous time wherein: W(t) for t≧0, with, W(0)=0 and such that the increment W(t)−W(s) is Gaussian with mean 0 and variance t−s for any 0≦s<t, and increments for non-overlapping time intervals are independent. Merton chose (somewhat arbitrarily) some maturity T and assumed that the company defaults provided that, at maturity, the company value is below the level of its debt D. In the case of no default, at maturity equity holders receive the excess amount of S=V−D, while debt holders receive the debt amount of D. In the case of default, equity holders receive nothing, while debt holders receive the amount of V. Thus, Merton views equity as the call option on the value of the company struck at its debt level, and debt as straight bond minus the put option on its value struck at the same level. As a result, the value of the company is the sum of its equity and debt.
The actual calculations in the Merton model are straightforward. For finding the value of equity and debt, we can use the usual Black-Scholes formulas:S=VN(d+)−e−rTDN(d−)  (2)δ=VN(−d+)+e−rTDN(d−)  (3)d±=[ln(V/D)+(r±σ2/2)T]/σ√{square root over (T)}  (4)wherein:    V: Value of the company    D: Debt at maturity.    r: Risk free interest rate.    σ: Standard deviation of the underlying asset, eg value of the company.    t Intermediate time between today (t=0) and maturity (t=T).    T: Maturity date.    S: Stock price, or equity    δ: Current debt value
Survival probability Q(t) is defined only for t=T:Q(T)=N(d−)  (5)
Here N(.) is the cumulative normal distribution. Equity option with maturity 0<T1<T and strike K can be priced as a compound option, and the corresponding equity volatility σS be determined by inverting the Black-Scholes formula. As a rule, Merton's model is calibrated to the market by choosing V and σ in such a way that the corresponding S and σS are matched exactly.
Using Merton's model in equation (1) above with the equity and debt values calculated using the Black-Scholes formulae (2), (3), (4) and (5), the probability of default is estimated from the market value of the equity, the face value of the debt and the volatility of the equity. The model is thus used to calculate an estimate of the likelihood of financial failure of a publicly traded company that is whether the debt position of a company will overrun its equity position, resulting in a financial failure.
Several commercial providers, including, for example, Moody-KMV, use Merton's model. Merton's model has obvious attractions because of its simplicity and intuitive character, but it does not address a number of very important questions, for instance, how to price bonds with different maturities in a consistent fashion. Merton's model further suffers from some fundamental mathematical inaccuracies relating to the default event. In particular, Merton's model inherently contemplates small changes, or ‘random walks’ in values from a starting point to a finishing point, i.e. from the current financial position of a publicly traded stock to a subsequent financial failure. The algorithm does not accommodate the large, random jumps often experienced in the marketplace. Further, the algorithm as initially set out contemplated “European style” option analysis, wherein options were exercised on a specified forward date. Merton's model does not contemplate a default before the due date. In practice, actions associated with financial failure can occur as a result of a single large event or a random walk on any given day. The reader is directed to the Internet websites    www.moodyskmv.com/about/index.html, and    http://www.moodyskmv.com/products/index.html,for publicly available information on the KMV credit risk analysis products and services.
At a later date, H. E. Leland modified Morton's model to restructure the description of a default event, thereby contemplating earlier default dates than the due date of the debt. Leland's extension of Merton's model assumes that the value of the company is governed by the standard risk-neutralized log-normal SDE, but that default happens whenever the value of the company falls below a certain continuously monitored constant barrier level LD, where L is the relative level of company's liabilities, or at maturity T when the value of the company falls below D. Leland uses the standard formulas for barrier options on lognormal assets in order to find the value of equity S, debt δ, and survival probabilities Q(t), Q(T):S=VN(d+)−e−rTDN(d−)−V(LD/V)2r/σ2+1N(ƒ+)+e−rTD(LD/V)2r/2−1N(ƒ−)  (6)δ=VN(−d+)+e−rTDN(d−)+V(LD/V)2r/σ2+1N(ƒ+)−e−rTD(LD/V)2r/σ2−1N(ƒ−)  (7)ƒ±=[ln(L2D/V)+(r±σ2/2)T]/σ√{square root over (T)}  (8)Q(t)=N(g+)−(LD/V)2r/σ2N(g−), t<T  (9)g±=[±ln(V/LD)+(r−σ2/2)t]/σ√{square root over (t)}  (10)Q(T)=N(d−)−(LD/V)2r/σ2−1N(ƒ−)  (11)where d± are defined by equation (4) and the remaining variables are as defined above.
Leland's model is rich enough for the user to be able to find the Credit Default Spread (CDS), i.e., the extra discount value provided for risky bonds as compared to non-risky bonds:
                              CDS          ⁡                      (            t            )                          =                              (                          1              -              R                        )                    ⁢                      (                                                            1                  -                                                            ⅇ                      rt                                        ⁢                                          Q                      ⁡                                              (                        t                        )                                                                                                                                  ∫                    0                    t                                    ⁢                                                            ⅇ                                              -                                                  rt                          ′                                                                                      ⁢                                          Q                      ⁡                                              (                                                  t                          ′                                                )                                                              ⁢                                                                                  ⁢                                          ⅆ                                              t                        ′                                                                                                        -              r                        )                                              (        12        )            where R is the recovery level for a particular debt seniority. While for medium and long maturities this spread is broadly comparable with the market, for short maturities it is very low, in sharp contrast with reality.
To rectify the above-mentioned problem, Credit Grades, a joint venture between JP Morgan, Deutsche Bank, and Goldman Sacks, proposed to make the recovery level L a normal random variable with the expected value of L and volatility ω. By doing so, they effectively replaced the calendar time t with the “shifted” time ζ=t+ ω2/σ2 and hence replaced the expression for the survival probability described above in equation (9) with the following:Q(t)=N(ĝ+)−(LD/V)2r/σ2N(ĝ−), t<T  (13)ĝ±=[±ln(V/LD)+(r−σ2/2)ζ]/σ√{square root over (ζ)}  (14)
As a result, they managed to increase the short-term CDSs, which still is given by equation (12). However, the model suffered in that it implicitly assumes that the value of the company can be below its default level from the very beginning, which, in the overwhelming majority of cases, is not true. Also, due to the approximate character of this formula, the initial survival probability Q(0) is less than one, which is mathematically (and financially) impossible. For further publicly available information the reader is directed to the Internet web page www.creditgrades.com/intro/intro.
In the later 1990's C. A. Zhou developed a jump-diffusion process, under which a company could default instantaneously because of a sudden drop in its value. The Zhou process was an extension of Leland's model, and addressed the issue of low CDSs in a more direct way. Zhou assumed that the value of the company is governed by an SDE with jumps:
                                          d            ⁢                                                  ⁢            V                    V                =                                            (                              r                -                                  λ                  ⁢                                                                          ⁢                  κ                                            )                        ⁢            dt                    +                      σ            ⁢                                                  ⁢            d            ⁢                                                  ⁢            W                    +                                    (                                                ⅇ                  J                                -                1                            )                        ⁢            d            ⁢                                                  ⁢            N                                              (        15        )            
Here N is the standard Poisson process with intensity λ, J is the lognormal jump size, which is assumed to be a random variable with a known probability density function φ(J), and κ is the expected value of the jump size,
  κ  =            ∫              -        ∞            ∞        ⁢                  (                              ⅇ            J                    -          1                )            ⁢              ϕ        ⁡                  (          J          )                    ⁢                          ⁢                        ⅆ          J                .            All other values are as defined above.
As in Leland's framework, default occurs if V(t) crosses the barrier LD, or, alternatively, if V(T)<D. The value of the company's equity S and its debt δ are computed by solving pricing problems for the corresponding barrier options. Due to the possibility of the company's value jumping down, CDSs for short maturities are significant. The key drawback of Zhou's model is its computational complexity. Also, its treatment of the recovery is somewhat ad hoc. It is very well known that barrier pricing problems for jump diffusions are very complicated. Zhou solves them via a Monte Carlo (MC) method. While MC methods are versatile, they are inaccurate and slow. Zhou's jump-diffusion model can match the size of credit spreads on corporate bonds and can generate various shapes of yield spread curves and marginal default rate curves, including upward-sloping, downward-sloping, at, and hump-shaped, even if the company is currently in good financial standing. The Zhou model also links recovery rates to company value at default in a natural way so that variation in recovery rates is endogenously generated in the model
Despite the improvements to the Merton's model offered by Leland and then Zhou, significant challenges relating to accuracy and operability exist with today's credit risk models. The present inventors believe that even with the Leland and Zhou improvements, the likelihood and timing of default is not accurately modeled. Further, many implementations of the various models are very difficult to program and execute on a computer.
Commercially available credit risk models exist which attempt to compensate for some of the deficiencies of described above.
The credit underlying securities pricing model, CUSP®, available from Credit Suisse First Boston™, is another commercially available credit risk model. CUSP® is an analytical model that relates an issuer's capital structure, stock price and the option implied volatility of its shares to credit risk. CUSP® provides systematic monitoring of credit risk from forward-looking, market-based measures (One Standard Deviation Spread Widening Risk (SWR)) and relative value tracking which incorporates both risk and return (Probability Weighted Return (PWR)). For further publicly available information, the reader is directed to the Internet website www.csfb.com/institutional/research/cusp.shtml.
While useful for estimating credit risks, the present inventors believe that the currently available algorithms and commercial models of the type described above still include inaccuracies in modeling both the random walks and the larger jumps experienced in the marketplace, as noted above. The more accurately such processes can be modeled, the higher the validity of a credit risk analysis tool. Not only is it desirable to accurately model such activities, but also the model itself must be able to be implemented, for example in a spreadsheet or other computer program, so as to practically operable.